Carrying out fault-tolerant topological quantum computation using non-Abelian anyons (e.g., Majorana zero modes) is a desired goal in quantum computation. However, the Gottesman-Knill theorem holds that if a system can only perform a certain subset of available quantum operations (e.g., operations from the Clifford group) in addition to the preparation and detection of qubit states in the computational basis, then that system is insufficient for universal quantum computation. Indeed, any measurement results in such a system could be reproduced within a local hidden variable theory, so that there is no need for a quantum mechanical explanation and therefore no possibility of quantum speedup.
Unfortunately, Clifford operations are precisely the ones available through braiding and measurement in systems supporting non-Abelian Majorana zero modes, which are otherwise a suitable candidate for topologically protected quantum computation. In order to move beyond the classically simulable subspace, an additional phase gate is desired. This phase gate would allow the system to violate the Bell-like CHSH inequality that would otherwise constrain a local hidden variable theory.